Problems with a "Fixed" Error Threshold (attn: John O'Sullivan)

I would just like to bring up an issue that arises in attempts to quantify and predict concordance in terms of error from JI, namely that of variations in sensitivity between different JI intervals.

I know John has been using 256/255 (around 7 cents, give or take) as a sort of "absolute" cut-off of what's acceptable, and I've heard other people mention 7 cents before as well as a good "rule of thumb". However, in my own experience, using a fixed error threshold like this is a mistake.

If you are using timbres with rich complements of harmonic partials, then JI ratios can be looked at as "the lowest set of coinciding partials" in the interval. So in a 7/6, the 7th partial of the lower tone coincides with the 6th partial of the higher tone. What this means is that more complex ratios will be more sensitive to a slight mistuning than to a gross mistuning, owing to the fact critical band roughness increases more quickly between higher partials than lower partials, and also decreases more quickly. For every 1 Hz that you shift a fundamental, its 7th partial changes by 7 Hz; same deal with all the other partials. Mistuning a 9/7 by 256/255 will produce a much greater level of discordance than mistuning a 5/4 by the same amount. Mistuning an 11/8 by that amount will be even more extreme. On the other hand, intervals like 5/4 and 6/5 can actually tolerate *more* than a 7-cent mistuning before roughness becomes a real problem.

Also, there is the "span effect" to take into account: when you get into intervals that span more than an octave, they become less sensitive to mistuning as well.

This is also assuming a pure harmonic timbre. Since real-world acoustic instruments often have a bit of inharmonicity to them, and their partials fluctuate a bit due to physical variations within the instrument and the environment and the impulse/attack used to produce sound, intervals don't always sound as in-tune or out-of-tune as we'd expect them to. Close to it, usually, but I have to say in my experience of playing EDOs on guitar that look out-of-tune on paper, the guitar seems to "forgive" error a lot more so than a synth (or even a sampled guitar).

Thus, a "fixed" error threshold will never be more than a rough "rule of thumb" and can often mislead one into either accepting intervals that are out-of-tune (listen to the 9/7 in 14-EDO, which is within the 256/255 threshold, and tell me if it really sounds in-tune), or ignoring intervals that sound in-tune (listen to the major 3rd in 16-EDO, which is well outside the 256/255 threshold, and tell me if it really sounds out-of-tune).

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> Thus, a "fixed" error threshold will never be more than a rough "rule of thumb" and can often mislead one into either accepting intervals that are out-of-tune (listen to the 9/7 in 14-EDO, which is within the 256/255 threshold, and tell me if it really sounds in-tune), or ignoring intervals that sound in-tune (listen to the major 3rd in 16-EDO, which is well outside the 256/255 threshold, and tell me if it really sounds out-of-tune).

We could always weight higher primes higher rather than lower, of course. Maybe I should start promoting Frobenius tuning. Graham got TE tuning going based on its mathematical advantages, and Frobenius tuning has even more of those. Most of anything as far as I can see.

Igs,

I did a few listening tests and here's what I got.

You said that mistuning a 9/7 by 256/255 will produce a much greater level of discordance than mistuning 5/4 by the same amount. The fact is that 9/7 is a much weaker interval than 5/4. So a perfectly tuned 9/7 will always sound much weaker than a tempered (say, 7 cents) 5/4.

You said that intervals like 5/4 and 6/5 can actually tolerate *more* than a 7 cent tuning. On my keyboard I tuned the E key to 0 cents and tuned the G# key to 386.3 cents (a "just" 5/4) and tuned the A key to 394.3 cents (tempering the 5/4 by +8 cents). I used a Church Organ voice on my keyboard as this seems to have the most ideal harmonic series (i.e. the frequencies of the partials are very close to x, 2x, 3x, 4x etc and the amplitudes of the partials are very close to y, y/2, y/3, y/4 etc).

I sustained the E key (0 cents) and alternated between the 386.3c (a just 5/4) and the 394.3c (5/4 +8 cents) keys and the latter definitely sounded a bit harsh to my ears. Next I tried a smaller tempering, 6 cents this time, and it sounded tolerable. So far my 6.776 cents tolerance seems ok.

You asked me if the 9/7 interval in 14EDO really sounds in tune. On first listen I thought no, it doesn't. But remember that 9/7 is a very weak interval anyway. Do you accept a 9/7 when it is justly tuned? I tuned my keyboard to play both a just 9/7 and a 14EDO 9/7 (6.5 cents flatter than just) and in this context the 14EDO 9/7 sounded acceptable to me.

Finally you asked me if a Major Third in 16EDO really sounds out of tune. You got me there. I have to admit it sounds tolerable even though it is flatter than just by 11.3 cents. Still, when comparing the 16EDO 5/4 to a just 5/4 there does seem to be a significant difference. So on this last point I'm not sure. I'll think about it.

Did you read my recent post (97470) on the good triads and tetrads that occur in 22EDO within 7.2 cents accuracy? 22 seems to me to be by far the best EDO less than 25.

John.

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> I would just like to bring up an issue that arises in attempts to quantify and predict concordance in terms of error from JI, namely that of variations in sensitivity between different JI intervals.
>
> I know John has been using 256/255 (around 7 cents, give or take) as a sort of "absolute" cut-off of what's acceptable, and I've heard other people mention 7 cents before as well as a good "rule of thumb". However, in my own experience, using a fixed error threshold like this is a mistake.
>
> If you are using timbres with rich complements of harmonic partials, then JI ratios can be looked at as "the lowest set of coinciding partials" in the interval. So in a 7/6, the 7th partial of the lower tone coincides with the 6th partial of the higher tone. What this means is that more complex ratios will be more sensitive to a slight mistuning than to a gross mistuning, owing to the fact critical band roughness increases more quickly between higher partials than lower partials, and also decreases more quickly. For every 1 Hz that you shift a fundamental, its 7th partial changes by 7 Hz; same deal with all the other partials. Mistuning a 9/7 by 256/255 will produce a much greater level of discordance than mistuning a 5/4 by the same amount. Mistuning an 11/8 by that amount will be even more extreme. On the other hand, intervals like 5/4 and 6/5 can actually tolerate *more* than a 7-cent mistuning before roughness becomes a real problem.
>
> Also, there is the "span effect" to take into account: when you get into intervals that span more than an octave, they become less sensitive to mistuning as well.
>
> This is also assuming a pure harmonic timbre. Since real-world acoustic instruments often have a bit of inharmonicity to them, and their partials fluctuate a bit due to physical variations within the instrument and the environment and the impulse/attack used to produce sound, intervals don't always sound as in-tune or out-of-tune as we'd expect them to. Close to it, usually, but I have to say in my experience of playing EDOs on guitar that look out-of-tune on paper, the guitar seems to "forgive" error a lot more so than a synth (or even a sampled guitar).
>
> Thus, a "fixed" error threshold will never be more than a rough "rule of thumb" and can often mislead one into either accepting intervals that are out-of-tune (listen to the 9/7 in 14-EDO, which is within the 256/255 threshold, and tell me if it really sounds in-tune), or ignoring intervals that sound in-tune (listen to the major 3rd in 16-EDO, which is well outside the 256/255 threshold, and tell me if it really sounds out-of-tune).
>
> -Igs
>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> We could always weight higher primes higher rather than lower, of course. Maybe I should
> start promoting Frobenius tuning. Graham got TE tuning going based on its mathematical
> advantages, and Frobenius tuning has even more of those. Most of anything as far as I can
> see.
>

That might help, but there's still the issue of cases where discordance can decrease as error increases past a certain threshold, even when you're still approximating the same intervals. 24-EDO treated as a 7-limit temperament is a great example of this. 0-700-950 is obviously a 4:6:7, but it's a lot more concordant than the ~20-cent error on 7/4 suggests it should be. Try playing 0-700-950, 0-700-958, 0-700-965, 0-700-970 and see what you notice. This will work best with harmonically-rich timbres, and won't work at all with pure sine waves.

-Igs

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:

> You said that mistuning a 9/7 by 256/255 will produce a much greater level of
> discordance than mistuning 5/4 by the same amount. The fact is that 9/7 is a much
> weaker interval than 5/4. So a perfectly tuned 9/7 will always sound much weaker than a > tempered (say, 7 cents) 5/4.

Yes, and that's one of the reasons that it can't tolerate nearly as much mistuning as a 5/4. Do you believe your 256/255 threshold still applies to 9/7?

> I sustained the E key (0 cents) and alternated between the 386.3c (a just 5/4) and the
> 394.3c (5/4 +8 cents) keys and the latter definitely sounded a bit harsh to my ears. Next > I tried a smaller tempering, 6 cents this time, and it sounded tolerable. So far my 6.776 > cents tolerance seems ok.

Try a 417-cent major 3rd on for size.

> You asked me if the 9/7 interval in 14EDO really sounds in tune. On first listen I thought > no, it doesn't. But remember that 9/7 is a very weak interval anyway. Do you accept a
> 9/7 when it is justly tuned? I tuned my keyboard to play both a just 9/7 and a 14EDO
> 9/7 (6.5 cents flatter than just) and in this context the 14EDO 9/7 sounded acceptable to > me.

9/7 when tuned Just is certainly acceptable. The one in 14 really seems to lack the coherence of the Just version. For another comparison, try a 9/7 of about 441 cents.

> Finally you asked me if a Major Third in 16EDO really sounds out of tune. You got me
> there. I have to admit it sounds tolerable even though it is flatter than just by 11.3 cents. > Still, when comparing the 16EDO 5/4 to a just 5/4 there does seem to be a significant
> difference. So on this last point I'm not sure. I'll think about it.

That's the funny thing about major 3rds. They seem to tolerate more error in the flat direction than the sharp direction. You should also compare 19-EDO's major 3rd, which is around 379 cents--just outside your threshold.

> Did you read my recent post (97470) on the good triads and tetrads that occur in 22EDO
> within 7.2 cents accuracy? 22 seems to me to be by far the best EDO less than 25.

Yes. 22 has long been acknowledged to be the closest to 11-limit JI you can get below 31-EDO. I've played in 22 extensively. I don't like it, but that's less to do with how its harmonies are tuned and more to do with the melodic/tonal structures that are available in 22. I just don't really like how chords "move" in 22. But I'm kind of an aberration in this regard.

-Igs

-Igs

Hey Igs,

Another thing to consider is that although higher partials are more sensitive to being mistuned, they are also weaker in amplitude. How much this effects the audible result surely depends on the timbre in question, too!

Best,
AKJ

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> I would just like to bring up an issue that arises in attempts to quantify and predict concordance in terms of error from JI, namely that of variations in sensitivity between different JI intervals.
>
> I know John has been using 256/255 (around 7 cents, give or take) as a sort of "absolute" cut-off of what's acceptable, and I've heard other people mention 7 cents before as well as a good "rule of thumb". However, in my own experience, using a fixed error threshold like this is a mistake.
>
> If you are using timbres with rich complements of harmonic partials, then JI ratios can be looked at as "the lowest set of coinciding partials" in the interval. So in a 7/6, the 7th partial of the lower tone coincides with the 6th partial of the higher tone. What this means is that more complex ratios will be more sensitive to a slight mistuning than to a gross mistuning, owing to the fact critical band roughness increases more quickly between higher partials than lower partials, and also decreases more quickly. For every 1 Hz that you shift a fundamental, its 7th partial changes by 7 Hz; same deal with all the other partials. Mistuning a 9/7 by 256/255 will produce a much greater level of discordance than mistuning a 5/4 by the same amount. Mistuning an 11/8 by that amount will be even more extreme. On the other hand, intervals like 5/4 and 6/5 can actually tolerate *more* than a 7-cent mistuning before roughness becomes a real problem.
>
> Also, there is the "span effect" to take into account: when you get into intervals that span more than an octave, they become less sensitive to mistuning as well.
>
> This is also assuming a pure harmonic timbre. Since real-world acoustic instruments often have a bit of inharmonicity to them, and their partials fluctuate a bit due to physical variations within the instrument and the environment and the impulse/attack used to produce sound, intervals don't always sound as in-tune or out-of-tune as we'd expect them to. Close to it, usually, but I have to say in my experience of playing EDOs on guitar that look out-of-tune on paper, the guitar seems to "forgive" error a lot more so than a synth (or even a sampled guitar).
>
> Thus, a "fixed" error threshold will never be more than a rough "rule of thumb" and can often mislead one into either accepting intervals that are out-of-tune (listen to the 9/7 in 14-EDO, which is within the 256/255 threshold, and tell me if it really sounds in-tune), or ignoring intervals that sound in-tune (listen to the major 3rd in 16-EDO, which is well outside the 256/255 threshold, and tell me if it really sounds out-of-tune).
>
> -Igs
>

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, "john777music" <jfos777@> wrote:
>
> > You said that mistuning a 9/7 by 256/255 will produce a much greater level of
> > discordance than mistuning 5/4 by the same amount. The fact is that 9/7 is a much
> > weaker interval than 5/4. So a perfectly tuned 9/7 will always sound much weaker than a > tempered (say, 7 cents) 5/4.
>
> Yes, and that's one of the reasons that it can't tolerate nearly as much mistuning as a 5/4. Do you believe your 256/255 threshold still applies to 9/7?

Yes I do, in my listening test the 14EDO 9/7 (6.5 cents flat of just) sounded acceptable when compared to a just 9/7.

>
> > I sustained the E key (0 cents) and alternated between the 386.3c (a just 5/4) and the
> > 394.3c (5/4 +8 cents) keys and the latter definitely sounded a bit harsh to my ears. Next > I tried a smaller tempering, 6 cents this time, and it sounded tolerable. So far my 6.776 > cents tolerance seems ok.
>
> Try a 417-cent major 3rd on for size.

Will do.

>
> > You asked me if the 9/7 interval in 14EDO really sounds in tune. On first listen I thought > no, it doesn't. But remember that 9/7 is a very weak interval anyway. Do you accept a
> > 9/7 when it is justly tuned? I tuned my keyboard to play both a just 9/7 and a 14EDO
> > 9/7 (6.5 cents flatter than just) and in this context the 14EDO 9/7 sounded acceptable to > me.
>
> 9/7 when tuned Just is certainly acceptable. The one in 14 really seems to lack the coherence of the Just version. For another comparison, try a 9/7 of about 441 cents.

Will do.

>
> > Finally you asked me if a Major Third in 16EDO really sounds out of tune. You got me
> > there. I have to admit it sounds tolerable even though it is flatter than just by 11.3 cents. > Still, when comparing the 16EDO 5/4 to a just 5/4 there does seem to be a significant
> > difference. So on this last point I'm not sure. I'll think about it.
>
> That's the funny thing about major 3rds. They seem to tolerate more error in the flat direction than the sharp direction.

That certainly seems to be the case and I intend to look into this.

You should also compare 19-EDO's major 3rd, which is around 379 cents--just outside your threshold.

Will do.

>
> > Did you read my recent post (97470) on the good triads and tetrads that occur in 22EDO
> > within 7.2 cents accuracy? 22 seems to me to be by far the best EDO less than 25.
>
> Yes. 22 has long been acknowledged to be the closest to 11-limit JI you can get below 31-EDO. I've played in 22 extensively. I don't like it, but that's less to do with how its harmonies are tuned and more to do with the melodic/tonal structures that are available in 22. I just don't really like how chords "move" in 22. But I'm kind of an aberration in this regard.
>
> -Igs
>
> -Igs
>

Can you give me an example of an interval where the greater the error the less the discordance? Sounds wrong to me but I'm curious.

John.

John (to Igs)>"You said that mistuning a 9/7 by 256/255 will produce a much greater
level of discordance than mistuning 5/4 by the same amount. The fact is
that 9/7 is a much weaker interval than 5/4. So a perfectly tuned 9/7
will always sound much weaker than a tempered (say, 7 cents) 5/4."

  I think Igs has a great point...higher limit intervals seem MUCH more sensitive to mis-tuning.  This may also explain why higher-limit intervals have smaller fields of attraction...even to the point Harmonic Entropy seems to ignore them.  It seems to put a lot of things together...

--- In tuning@yahoogroups.com, "akjmicro" <aaron@...> wrote:
>
> Hey Igs,
>
> Another thing to consider is that although higher partials are more sensitive to being
> mistuned, they are also weaker in amplitude. How much this effects the audible result
> surely depends on the timbre in question, too!

Right, of course. But that's why I always test concordance with sawtooth waves. To me, the whole point of using Just or near-Just intervals is that they remain coherent under saturated gain situations (like using a fuzz pedal on guitar). There's no real point in playing in JI if you're using timbres where the higher partials are weak. On an acoustic guitar, 12-TET is barely distinguishable from JI. This is why Jon Catler does all his demos with an overdriven guitar rather than a clean one--if it's clean, the difference is too subtle. Even on a piano, the difference is subtle enough to pass most listeners' ears unnoticed (unless you're using non-12ish intervals, of course). I don't know if you're a Radiohead fan, but there's a song on their new album called "Codex" which I would swear is in JI, except that I'm sure it's not.

Of course, beyond critical band roughness, combination tones are also important in high-gain situations, and I haven't even *begun* to study how those are affected by tempering.

-Igs

On Saturday, April 2, 2011, cityoftheasleep <igliashon@...> wrote:
>
> If you are using timbres with rich complements of harmonic partials, then JI ratios can be looked at as "the lowest set of coinciding partials" in the interval. So in a 7/6, the 7th partial of the lower tone coincides with the 6th partial of the higher tone. What this means is that more complex ratios will be more sensitive to a slight mistuning than to a gross mistuning, owing to the fact critical band roughness increases more quickly between higher partials than lower partials, and also decreases more quickly.

True, but keep in mind that higher partials also have less energy that
lower partials, so the beating is less audible. This is why a major
third that's 14 cents sharp, as in that of 12-tet, is more tolerable
than a 3/2 that's 14 cents sharp. On the other hand, higher intervals
are also more sensitive to mistuning in that they tend to suffer more
from what Carl recently called the SPAN effect; you can sharpen a 3/2
by 30 cents and it'll still be recognizable as a 3/2, but sharpen 8/7
by 30 cents and you now have a 7/6.

There's also the fact that there's a maximum point of critical band
dissonance that once you get past, doesn't sound so bad.

--
-Mike

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:

> Can you give me an example of an interval where the greater the error the less the
> discordance? Sounds wrong to me but I'm curious.

The major 3rd is a good example: start from a Just 5/4. Then try 395, 400, 405, 410, 415, and 420 cents. Somewhere between 410 and 420 cents, I hear the discordance diminish, so the peak of discordance is somewhere between 395 and 405. You can also try the perfect fifth: start from 3/2, then flatten to 695, 690, 685, 680, 675, 670, 665. The discordance seems to peak between 680 and 675, and then starts to diminish a bit, bottoming out around 665 cents and then climbing back up as you approach 650 cents.

Of course, this effect is most noticeable with timbres where at least the first 16 partials have equal power.

-Igs

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>   I think Igs has a great point...higher limit intervals seem MUCH more sensitive to mis-
> tuning.  This may also explain why higher-limit intervals have smaller fields of
> attraction...even to the point Harmonic Entropy seems to ignore them.  It seems to put a lot
> of things together...

There's also the fact that they are more complex in terms of both combination tones and periodicity. But with harmonically rich timbres, you can sometimes tell when partials become coincident. Even Paul Erlich admitted to me that at 17/13, some amount of beating decreases from the irrational areas near by.

To me the upshot of it all is that thinking in terms of ratios and concordance is more of a headache than it's worth. My final resolution, after learning as much psychoacoustic theory that I can stand, is that as long as I'm working with EDOs--where the pitches are already decided for me--the only guide that is useful or relevant to my practice of music remains my ears. Thinking in terms of JI and error and all that has only sown confusion and frustration for me, and has never actually informed or improved my playing or composition. So I'm letting it go.

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> Thinking in terms of JI and error and all that has only sown confusion and frustration for me, and has never actually informed or improved my playing or composition. So I'm letting it go.

That's because you are trying to apply it to to high-error temperaments.

Igs>"Somewhere between 410 and 420 cents, I hear the discordance diminish"

    No kidding...by 420 cents, you are approaching a field of attraction of 9/7.  

>"You can also try the perfect fifth: start from 3/2, then flatten to 695,
690, 685, 680, 675, 670, 665. The discordance seems to peak between
680 and 675,   "
 
   Coincidentally, about 680 cents is dead in between 22/15 and 3/2.  Guess what...I have long suspected 22/15 has it's own field of attraction.

   What I'm gaining from Igs's theory about partials, indirectly, is that it proves higher-limit JI does, by nature, have fields of attraction, just narrower fields than lower-limit rational intervals. It seems to be a consistent pattern in the results...

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> What I'm gaining from Igs's theory about partials, indirectly, is that it proves higher-limit JI > does, by nature, have fields of attraction, just narrower fields than lower-limit rational
> intervals. It seems to be a consistent pattern in the results...

For one, this isn't "my theory". This is straight out of Sethares. And for the record, I don't think this is necessarily about fields of attraction of higher-limit intervals. If anything, it seems to suggest what Cameron was calling "asonance"--intervals where the significant partials of the two notes are all out of each other's critical bands but, none are coincident. It's not the same thing as concordance, really. There is still plenty of beating happening, but it's all toned down just enough so as to be less irritating. Nothing "locks in" the way it does at a Just interval, and it's impossible (or at least pointless) to describe these points with an exact identity. The drop in discordance is quite small compared to a Just interval.

-Igs

Hi Igs,

I've been testing your ideas and got some interesting results. You said that the 5/4 can tolerate more mistuning when you flatten it than if you sharpen it. You were right.

I started with 9/7, a weak interval. I tempered it by +6.5 cents and it still sounded okay. Then I raised it by +8 cents and it sounded sour (so my 6.776 cents "upper" threshold seems good so far). Going the other way (flattening the 9/7) I found that I could flatten the 9/7 by as much as 27 cents and it still sounded "clean". Flattening by 28 cents or more the 9/7 sounded sour.

You said that sometimes the greater the error, the less the discordance and you were right again. I noticed when I was progressively flattening the 9/7 that the beating (discordance) did indeed seem to lessen and the flattened intervals sounded smoother. On the other hand the intervals that were closer to "just" sounded more "resolved". My gut feeling (perhaps because the math looks cuter) is that I would choose "more resolved" over "less beating".

Next I tried 1/1. I tempered the 1/1 by +6.7 cents and it sounded good. Next I raised it by +8 cents and it definitely sounded sour. So again my 6.776 cents "upper" threshold seems good.

Next I tried 2/1. I raised it by 6.7 cents and it was good. +8 cents and it was sour. I flattened 2/1 by -26 cents and the beating was *very* obvious but not "nasty". I tried a -29 cents tempering and the beating began to sound nasty. In this case the -27 cents tolerance is questionable but I wouldn't use a tempered octave in a scale so maybe it doesn't matter.

Next I tried 5/4 and again, flattening the interval by 27 cents, it still sounded "clean". Next I tried -29 cents and I didn't like it.

By coincidence 64/63 is between 27 cents and 28 cents. This is similar to 256/255 (i.e. 2^N/(2^N - 1). So if I had to choose precise values for tempering I would choose +6.776 cents (256/255) for the upper tolerance and -27.264 cents (64/63) for the lower tolerance. I can't prove that these precise values are correct but they look cute and seem reasonable when tested.

John.

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, "john777music" <jfos777@> wrote:
>
> > Can you give me an example of an interval where the greater the error the less the
> > discordance? Sounds wrong to me but I'm curious.
>
> The major 3rd is a good example: start from a Just 5/4. Then try 395, 400, 405, 410, 415, and 420 cents. Somewhere between 410 and 420 cents, I hear the discordance diminish, so the peak of discordance is somewhere between 395 and 405. You can also try the perfect fifth: start from 3/2, then flatten to 695, 690, 685, 680, 675, 670, 665. The discordance seems to peak between 680 and 675, and then starts to diminish a bit, bottoming out around 665 cents and then climbing back up as you approach 650 cents.
>
> Of course, this effect is most noticeable with timbres where at least the first 16 partials have equal power.
>
> -Igs
>

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:

> I found that I could flatten the 9/7 by as much as 27 cents and it still sounded "clean". Flattening by 28 cents or more the 9/7 sounded sour.

If you flatten 9/7 by 99/98, which is 17.6 cents, you get 14/11, so obviously you can't take this analysis that far. The mediant between 9/7 and 14/11 is 23/18, which is 424.4 cents. I wouldn't push discussion of 9/7 past there. (9/7)/(23/18) = 162/161, which is 10.7 cents, by the way.

pered octave in a scale so maybe it doesn't matter.
>
> Next I tried 5/4 and again, flattening the interval by 27 cents, it still sounded "clean". Next I tried -29 cents and I didn't like it.
>
> By coincidence 64/63 is between 27 cents and 28 cents.

More to the point, 65/64 is 26.8 cents, and flattening 5/4 by that amount gives you 16/13, so it seems to me you are talking about 16/13 and a little under, with 16/13 being OK in your book, but you don't want it too much flattened.

> This is similar to 256/255 (i.e. 2^N/(2^N - 1). So if I had to choose precise values for tempering I would choose +6.776 cents (256/255) for the upper tolerance and -27.264 cents (64/63) for the lower tolerance. I can't prove that these precise values are correct but they look cute and seem reasonable when tested.

If powers of 2 are so great why is it that 2^n generally sucks as an edo? (Now I suppose I'll hear from fans of 16.)

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
> If you flatten 9/7 by 99/98, which is 17.6 cents, you get 14/11, so obviously you can't take this analysis that far. The mediant between 9/7 and 14/11 is 23/18, which is 424.4 cents. I wouldn't push discussion of 9/7 past there. (9/7)/(23/18) = 162/161, which is 10.7 cents, by the way.
>

You're assuming that 14/11 is Just. I don't think, according to HE or Tenney height, that that claim is supported.

> If powers of 2 are so great why is it that 2^n generally sucks as an edo? (Now I suppose I'll hear from fans of 16.)
>

LOL.

-Igs

Igs>"You're assuming that 14/11 is Just. I don't think, according to HE or Tenney height, that that claim is supported."

   To my ears, Tenney Height is a load of garbage for any rational fraction with a Tenney Height of around 70 or over.  So I'm guessing you'd say 11/7, 13/9,  11/9, 16/9, 12/7....have no place as "Just intervals".

   It seems obvious to me when we use Tenney Height we are dodging a major problem, which is how to rate 9, 11...or higher limit intervals reliably.  Tenney Height does not cut it, and perhaps you can argue John's rating of fractions does not cut it (provided you give examples).  But if John's doesn't cut it (along with Tenney Height)...here's the real question: what can you come up with that does?

> If powers of 2 are so great why is it that 2^n generally sucks as an edo? (Now I suppose I'll hear from fans of 16.)

  This also seems to be side-tracking on a tangent from the original problem.  Of course 2 as an exponential division of a scale IE in an EDO is going to function differently than a power of 2 in a fraction. 

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
>
> Hi Igs,
> I've been testing your ideas and got some interesting results.
> You said that the 5/4 can tolerate more mistuning when you
> flatten it than if you sharpen it. You were right.

For my part, I'll point out that this claim is blatantly false.
It sounds different sharp than flat, but not worse. Maybe you
prefer one or the other today, or next week, but such conclusions
are likely due to experimental bias, or personal preference.

Igs, I thought we'd settled this offlist.

-Carl

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

> > If powers of 2 are so great why is it that 2^n generally sucks as an edo? (Now I suppose I'll hear from fans of 16.)
>
>   This also seems to be side-tracking on a tangent from the original problem.  Of course 2 as an exponential division of a scale IE in an EDO is going to function differently than a power of 2 in a fraction. 

Numerology is numerology wherever you find it.

...Ok Gene, then what's the alternative?  IE how can we rate ratios with a Tenney Height of over 70 with fair reliability?

--- On Sun, 4/3/11, genewardsmith <genewardsmith@...> wrote:

From: genewardsmith <genewardsmith@...>
Subject: [tuning] Re: Problems with a "Fixed" Error Threshold (attn: John O'Sullivan)
To: tuning@yahoogroups.com
Date: Sunday, April 3, 2011, 2:59 PM

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

> > If powers of 2 are so great why is it that 2^n generally sucks as an edo? (Now I suppose I'll hear from fans of 16.)

>

>   This also seems to be side-tracking on a tangent from the original problem.  Of course 2 as an exponential division of a scale IE in an EDO is going to function differently than a power of 2 in a fraction. 

Numerology is numerology wherever you find it.

> You're assuming that 14/11 is Just. I don't think, according to HE or Tenney height,
> that that claim is supported.

I just used it as part of a 28:33:42 chord in the Cantonese Bagatelle. It sounds pretty good there. After listening to the canton scale that Gene put together a while back, I've got to say I don't mind the 11-limit stuff in it at all.

Regards,
Jake

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> >
> >
> > If you flatten 9/7 by 99/98, which is 17.6 cents, you get 14/11, so obviously you can't take this analysis that far. The mediant between 9/7 and 14/11 is 23/18, which is 424.4 cents. I wouldn't push discussion of 9/7 past there. (9/7)/(23/18) = 162/161, which is 10.7 cents, by the way.
> >
>
> You're assuming that 14/11 is Just. I don't think, according to HE or Tenney height, that that claim is supported.

I'm assuming no such thing, and the claim any such conclusion could in any case be drawn from Tenney height is nonsense unless you somehow know what the consonance limit is. Do you? If so, how and what is it? I am assuming it's audible, which is true.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> ...Ok Gene, then what's the alternative?  IE how can we rate ratios with a Tenney Height of over 70 with fair reliability?

Try a listening test.

Carl>"For my part, I'll point out that this claim is blatantly false.  It sounds different sharp than flat, but not worse. Maybe you prefer one or the other today, or next week, but such conclusions are likely due to experimental bias, or personal preference."

   Why, because Carl says so?!   Sounds like a double standard to me: apparently no one else can make sweeping accusations without explanations...except Carl?!  Where is your evidence?

   John, myself, Igs (apparently) and others appear to agree that sharpness VERY OFTEN is more tolerable than flatness.  If it were just purely "personal preference"...certainly only one person would agree...

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

>    John, myself, Igs (apparently) and others appear to agree that sharpness VERY OFTEN is more tolerable than flatness.  If it were just purely "personal preference"...certainly only one person would agree...
>

I tend to like a sharp fifth better than a flat one, but I don't think it is one bit more tolerable. My vote goes to Carl. Of course, here's another listening test candidate.

Me> ...Ok Gene, then what's the alternative?  IE how can we rate ratios with a Tenney Height of over 70 with fair reliability?

Gene>Try a listening test.

   Right...which seems to lead back to "Personal preferences" IE "Did what I just hear sound good to me?"  So if we are treading in an area that no prominent theory solves, at the very least, I figure we should be open to testing "unscientific" personal preferences in that area until we see a pattern in listening test results...and certainly not gun-down any "personal theories" before we've had time to test them.

Me>    John, myself, Igs (apparently) and others appear to agree that
sharpness VERY OFTEN is more tolerable than flatness.  If it were just
purely "personal preference"...certainly only one person would agree...

Gene>"I tend to like a sharp fifth better than a flat one, but I don't think
it is one bit more tolerable. My vote goes to Carl. Of course, here's
another listening test candidate."

    Right, so apparently...the score is 3 to 2 so far...not too far from equal but, go figure...we almost certainly need more testers to really get a reliable impression.  Anyone else game?

--- On Sun, 4/3/11, genewardsmith <genewardsmith@...> wrote:

From: genewardsmith <genewardsmith@...>
Subject: [tuning] Re: Problems with a "Fixed" Error Threshold (attn: John O'Sullivan)
To: tuning@yahoogroups.com
Date: Sunday, April 3, 2011, 3:10 PM

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

>

I tend to like a sharp fifth better than a flat one, but I don't think it is one bit more tolerable. My vote goes to Carl. Of course, here's another listening test candidate.

> Gene>"I tend to like a sharp fifth better than a flat one, but
> I don't think it is one bit more tolerable. My vote goes to
> Carl. Of course, here's another listening test candidate."
>
> Right, so apparently...the score is 3 to 2 so far...not
> too far from equal but, go figure...we almost certainly need
> more testers to really get a reliable impression. Anyone else
> game?

To begin to start thinking about letting it cross our minds that we're plausibly considering being "scientific", we'd need a *lot* of people to do the listening test -- and this list is probably the worst set of people to choose from, because it's full of people who self-selected for having an interest in different qualities of intervals!

Jake>"To begin to start thinking about letting it cross our minds that we're plausibly considering being "scientific", we'd need a *lot* of people to do the listening test -- and this list is probably the worst set of people to choose from, because it's full of people who self-selected for having an interest in different qualities of intervals!"

    Good point...which would seem to bring us to the task of testing a large group of people in the "general public" IE people who have virtually no trained interest in qualities of certain types of intervals and only a gut sense of more/less relaxed far as sounds/dyads.  Which is something I have just about been burned at the stake for: the idea of testing the "general public" rather than "experienced listeners"...but something I am still all for.  Any ideas how, where to do such a "global test"?  I assume the "Mechanical Turk" is out of bounds...

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:

> To begin to start thinking about letting it cross our minds that
> we're plausibly considering being "scientific", we'd need a *lot*
> of people to do the listening test -- and this list is probably
> the worst set of people to choose from, because it's full of
> people who self-selected for having an interest in different
> qualities of intervals!

Regarding 5:4 in particular, there's an enormous body of literature
on wide ones going back over 200 years to draw on.

Regarding tuning error pain asymmetry, this has been intensely
studied here over the years. It's not like I'm making this
stuff up.

-Carl

> Which is something I have just about been burned at the stake
> for: the idea of testing the "general public" rather than
> "experienced listeners"...but something I am still all for.

It depends on what you want, and what questions you're asking. If you're trying to get the opinions of "experienced listeners", then that's whom you should be polling. If you're trying to get general principles that apply to more people, you should poll more people.

Re: What You Want, I think that you'd want more people in this case, because we appear to be looking for something general. To make an analogy: Some political pollsters poll a large random sample of "eligible voters" or just "adults", because they want to know sentiment in the populace; others poll a large random sample of likely voters, because they want to see how the vote is going to turn out. I think we're trying to find out fundamental things about human beings, not things that human beings *can* achieve through ear training.

Re: What Questions You're Asking, I would stay focused on intervals heard in a variety of timbres. Asking who prefers Bach over Beethoven would be skewed by people who listen to Britney Spears; not that there couldn't be differences, but those differences wouldn't be that important to me. On the other hand, I would be interested in asking, "Which sounds better, this interval or that one?" with a 300-cent m3 and a 13/11 m3, and I'd ask that question of listeners everything, from Bach to Britney to Bachir.

I certainly don't think you'll discover anything all that interesting by having a sample of "people who listen to, write, and theorize about microtonal music". If this list is any indicator, mostly you'll just start a pissing contest. :)

> Any ideas how, where to do such a "global test"?

I would suggest:

1. Working with one or more colleges, getting people to do a random listening test and recording their opinions on a Web site set up for the purpose. Someone working for a Psych degree, perhaps -- not a music degree.

2. If anyone has connections with a classical radio station, find a DJ who is friendly to unusual music -- I'm thinking Terrence McKnight of WQXR in New York City, although I don't know him personally -- and a marketing manager who is focused on social media, and have them host a poll on their blog. Usually these stations can embed the MP3s right in the Web page, so no downloads are needed, and you can have people rate a whole bunch of specific intervals on a single page.

3. Set up a Web page and start a Twitter and / or Facebook poll. Send out the link with a few hashtags on it (e.g., #music or #science or #musictheory or whatever looks popular). Keep doing that for a few months or a year or forever. I can't set it up, but I would certainly tweet it out to my followers and share it with my Facebook friends.

Regards,
Jake

Carl said:

> Regarding 5:4 in particular, there's an enormous body of literature
> on wide ones going back over 200 years to draw on.

Super. I'm happy to use what people have already done. But when people are talking about 3-to-2 polls on this forum, that's about as far from "scientific" as we can get.

> Regarding tuning error pain asymmetry, this has been intensely
> studied here over the years. It's not like I'm making this
> stuff up.

Good. I haven't seen it, and it's coming up as new and nobody's talking about prior research. And, when people are talking about 3-to-2 polls on this forum...

Regards,
Jake

On Sun, Apr 3, 2011 at 11:07 AM, cityoftheasleep
<igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
> >
> > What I'm gaining from Igs's theory about partials, indirectly, is that it proves higher-limit JI > does, by nature, have fields of attraction, just narrower fields than lower-limit rational
> > intervals. It seems to be a consistent pattern in the results...
>
> For one, this isn't "my theory". This is straight out of Sethares.

Sethares doesn't mention that the energy decreases with each higher
partial? I'm sure he mentions that plenty of times.

-Mike

Jake>"If you're trying to get general principles that apply to more people, you should poll more people."

   Yes, that's what I'm trying for.  Hence when I listed a poll I said we need a lot more data from a general population.  The reason I'm polling this list...is because it's just about the only resource I have for such tests...not because I think it's an ideal source.

>"If this list is any indicator, mostly you'll just  start a pissing contest. :)"
   Argh....true that! :-S -> :-D

>"1. Working with one or more colleges, getting people to do a random listening test and recording their opinions on a Web site set up for the purpose. Someone working for a Psych degree, perhaps -- not a music degree."

Sounds good, now I just need to find said kind of person (without background bias)...

>"2. If anyone has connections with a classical radio station, find a DJ who is friendly to unusual music -- I'm thinking Terrence McKnight of WQXR in New York City, although I don't know him personally -- and a marketing manager who is focused on social media, and have them host a poll on their blog."

Great idea...I had no clue many of such DJs existed...I knew of only one into avant garde music, but he just fired from a local station in Houston, Texas...sadly.

>"3. Set up a Web page and start a Twitter and / or Facebook poll. Send out the link with a few hashtags on it (e.g., #music or #science or #musictheory or whatever looks popular). Keep doing that for a few months or a year or forever. "

    Have tried, but haven't had any luck getting people to show up.  Hmm...

--- On Sun, 4/3/11, Jake Freivald <jdfreivald@...> wrote:

From: Jake Freivald <jdfreivald@...>
Subject: Re: [tuning] Re: Problems with a "Fixed" Error Threshold (attn: John O'Sullivan)
To: tuning@yahoogroups.com
Date: Sunday, April 3, 2011, 4:06 PM

> Which is something I have just about been burned at the stake

> for: the idea of testing the "general public" rather than

> "experienced listeners"...but something I am still all for.

It depends on what you want, and what questions you're asking. If you're

trying to get the opinions of "experienced listeners", then that's whom

you should be polling. If you're trying to get general principles that

apply to more people, you should poll more people.

Re: What You Want, I think that you'd want more people in this case,

because we appear to be looking for something general. To make an

analogy: Some political pollsters poll a large random sample of

"eligible voters" or just "adults", because they want to know sentiment

in the populace; others poll a large random sample of likely voters,

because they want to see how the vote is going to turn out. I think

we're trying to find out fundamental things about human beings, not

things that human beings *can* achieve through ear training.

Re: What Questions You're Asking, I would stay focused on intervals

heard in a variety of timbres. Asking who prefers Bach over Beethoven

would be skewed by people who listen to Britney Spears; not that there

couldn't be differences, but those differences wouldn't be that

important to me. On the other hand, I would be interested in asking,

"Which sounds better, this interval or that one?" with a 300-cent m3 and

a 13/11 m3, and I'd ask that question of listeners everything, from Bach

to Britney to Bachir.

I certainly don't think you'll discover anything all that interesting by

having a sample of "people who listen to, write, and theorize about

microtonal music". If this list is any indicator, mostly you'll just

start a pissing contest. :)

> Any ideas how, where to do such a "global test"?

I would suggest:

1. Working with one or more colleges, getting people to do a random

listening test and recording their opinions on a Web site set up for the

purpose. Someone working for a Psych degree, perhaps -- not a music degree.

2. If anyone has connections with a classical radio station, find a DJ

who is friendly to unusual music -- I'm thinking Terrence McKnight of

WQXR in New York City, although I don't know him personally -- and a

marketing manager who is focused on social media, and have them host a

poll on their blog. Usually these stations can embed the MP3s right in

the Web page, so no downloads are needed, and you can have people rate a

whole bunch of specific intervals on a single page.

3. Set up a Web page and start a Twitter and / or Facebook poll. Send

out the link with a few hashtags on it (e.g., #music or #science or

#musictheory or whatever looks popular). Keep doing that for a few

months or a year or forever. I can't set it up, but I would certainly

tweet it out to my followers and share it with my Facebook friends.

Regards,

Jake

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:

> On the other hand, I would be interested in asking, "Which
> sounds better, this interval or that one?" with a 300-cent
> m3 and a 13/11 m3, and I'd ask that question of listeners
> everything, from Bach to Britney to Bachir.

As long as we're suggesting experiment designs here, I'll
point out that any cohort you put together is going to have
the shared bias of 12-ET. Familiarity/novelty can be confused
with tolerability by naive listeners.

One way around this involves comparing triads and larger
chords (where the sizes of individual intervals are less
apparent and may vary with respect to 12-ET in more than one
way at a time) and comparing the results to those of a
symmetric error measure, such as RMS error. (It turns
out that symmetric error works.)

It's also true that neither traditional sensory consonance
models, nor harmonic entropy, predict any significant
deviation from symmetric error.

-Carl

Carl>"Regarding tuning error pain asymmetry, this has been intensely studied here over the years. It's not like I'm making this stuff up."

    Interesting...so where can I find the papers on it?  Honestly I believe you, but I haven't heard of any papers on that topic and don't know where to look for them.

Carl>"As long as we're suggesting experiment designs here, I'll point out that any cohort you put together is going to have the shared bias of 12-ET. "

Good point.  As such wouldn't it make sense to use intervals WITHOUT any near equivalents in 12TET so nothing gets the "familiarity advantage"?

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> Igs, I thought we'd settled this offlist.

I dropped the matter, but for me it's hardly settled. And that's fine. After long discourses with Paul, he's persuaded me that for my purposes, I'm better off ignoring quantification and staying as far away from psychoacoustics as possible. He reminded me that I've not really taken such matters into account in making my own music, and that most musicians are unlikely to be persuaded by or concerned with arguments based on psychoacoustics.

At any rate, I recall we did agree on 19-EDO's 4:5:6:7 sounding more "septimal" than 22's, despite being slightly more discordant. I think with tempered 5/4's, the flatter ones maintain a bit more of the "quintal" ("pental"?) character--that "smooth, mellow, relaxed" sort of flavor. That's all I can really say on the matter. I think in the long run, mood is more important than concordance when tempering. But that's just me.

-Igs

My suggestion would be to convince someone to make the poll their
school project. Middle school science class and up would be fair game.
Set up a web site for the kid and let him or her have at it.
Got a nephew or niece?

Chris

On Sun, Apr 3, 2011 at 7:52 PM, Michael <djtrancendance@...> wrote:
>
>
>
>
> >"3. Set up a Web page and start a Twitter and / or Facebook poll. Send out the link with a few hashtags on it (e.g., #music or #science or #musictheory or whatever looks popular). Keep doing that for a few months or a year or forever. "
>
>     Have tried, but haven't had any luck getting people to show up.  Hmm...
>
>
>
>
>
>
> --- On Sun, 4/3/11, Jake Freivald <jdfreivald@...> wrote:
>
> From: Jake Freivald <jdfreivald@...>
> Subject: Re: [tuning] Re: Problems with a "Fixed" Error Threshold (attn: John O'Sullivan)
> To: tuning@yahoogroups.com
> Date: Sunday, April 3, 2011, 4:06 PM
>
>
>
> > Which is something I have just about been burned at the stake
> > for: the idea of testing the "general public" rather than
> > "experienced listeners"...but something I am still all for.
>
> It depends on what you want, and what questions you're asking. If you're
> trying to get the opinions of "experienced listeners", then that's whom
> you should be polling. If you're trying to get general principles that
> apply to more people, you should poll more people.
>
> Re: What You Want, I think that you'd want more people in this case,
> because we appear to be looking for something general. To make an
> analogy: Some political pollsters poll a large random sample of
> "eligible voters" or just "adults", because they want to know sentiment
> in the populace; others poll a large random sample of likely voters,
> because they want to see how the vote is going to turn out. I think
> we're trying to find out fundamental things about human beings, not
> things that human beings *can* achieve through ear training.
>
> Re: What Questions You're Asking, I would stay focused on intervals
> heard in a variety of timbres. Asking who prefers Bach over Beethoven
> would be skewed by people who listen to Britney Spears; not that there
> couldn't be differences, but those differences wouldn't be that
> important to me. On the other hand, I would be interested in asking,
> "Which sounds better, this interval or that one?" with a 300-cent m3 and
> a 13/11 m3, and I'd ask that question of listeners everything, from Bach
> to Britney to Bachir.
>
> I certainly don't think you'll discover anything all that interesting by
> having a sample of "people who listen to, write, and theorize about
> microtonal music". If this list is any indicator, mostly you'll just
> start a pissing contest. :)
>
> > Any ideas how, where to do such a "global test"?
>
> I would suggest:
>
> 1. Working with one or more colleges, getting people to do a random
> listening test and recording their opinions on a Web site set up for the
> purpose. Someone working for a Psych degree, perhaps -- not a music degree.
>
> 2. If anyone has connections with a classical radio station, find a DJ
> who is friendly to unusual music -- I'm thinking Terrence McKnight of
> WQXR in New York City, although I don't know him personally -- and a
> marketing manager who is focused on social media, and have them host a
> poll on their blog. Usually these stations can embed the MP3s right in
> the Web page, so no downloads are needed, and you can have people rate a
> whole bunch of specific intervals on a single page.
>
> 3. Set up a Web page and start a Twitter and / or Facebook poll. Send
> out the link with a few hashtags on it (e.g., #music or #science or
> #musictheory or whatever looks popular). Keep doing that for a few
> months or a year or forever. I can't set it up, but I would certainly
> tweet it out to my followers and share it with my Facebook friends.
>
> Regards,
> Jake
>
>

--- "cityoftheasleep" <igliashon@...> wrote:

> I dropped the matter, but for me it's hardly settled.

I guess I'm still waiting for your reply to:

>>> Are you really so certain that, say, an interval 5 cents
>>> flat will always sound rougher than if it were 2 cents sharp?
>>
>> Yup. Can you find a counterexample?

> At any rate, I recall we did agree on 19-EDO's 4:5:6:7
> sounding more "septimal" than 22's, despite being slightly
> more discordant. I think with tempered 5/4's, the flatter
> ones maintain a bit more of the "quintal" ("pental"?)
> character--that "smooth, mellow, relaxed" sort of flavor.
> That's all I can really say on the matter. I think in the
> long run, mood is more important than concordance when
> tempering. But that's just me.

If these moods exist (are shared among the general population)
then they include / are influenced by concordance.

-Carl

Chris>"Middle school science class and up would be fair game. Set up a web site for the kid and let him or her have at it. Got a nephew or niece?"

Well, it's a twisted, but ingenious idea. Picking younger kids virtually guarantees personal bias toward certain intervals (often via "numerology") is eliminated.

I don't have niece/nephew yet...at this rate my up and coming daughter will be in middle school before that happens. But hey...I could always go on a sidetrack and find someone online willing to do it (IE a middle school student looking for class project ideas).

A sharp 3:2 will give a you flat 4:3, a low 5:4 a high 8:5 (assuming an untempered 2 and that we're not using some bizarre system in which these intervals are not mapped in a modulo 2 inverse relation)

Were we to insist that either sharper or flatter is the "better" direction of temperament, consistency would demand that we temper the octave.

As far as polling: cultured opinions are not physiological facts.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Me>    John, myself, Igs (apparently) and others appear to agree that
> sharpness VERY OFTEN is more tolerable than flatness.  If it were just
> purely "personal preference"...certainly only one person would agree...
>
>
> Gene>"I tend to like a sharp fifth better than a flat one, but I don't think
> it is one bit more tolerable. My vote goes to Carl. Of course, here's
> another listening test candidate."
>
>     Right, so apparently...the score is 3 to 2 so far...not too far from equal but, go figure...we almost certainly need more testers to really get a reliable impression.  Anyone else game?
>
> --- On Sun, 4/3/11, genewardsmith <genewardsmith@...> wrote:
>
> From: genewardsmith <genewardsmith@...>
> Subject: [tuning] Re: Problems with a "Fixed" Error Threshold (attn: John O'Sullivan)
> To: tuning@yahoogroups.com
> Date: Sunday, April 3, 2011, 3:10 PM
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>  
>
>
>
>
>
>
>
>
>
>
>
>
>
> --- In tuning@yahoogroups.com, Michael <djtrancendance@> wrote:
>
>
> >
>
>
>
> I tend to like a sharp fifth better than a flat one, but I don't think it is one bit more tolerable. My vote goes to Carl. Of course, here's another listening test candidate.
>

On Mon, Apr 4, 2011 at 1:20 AM, lobawad <lobawad@...> wrote:
>
> A sharp 3:2 will give a you flat 4:3, a low 5:4 a high 8:5 (assuming an untempered 2 and that we're not using some bizarre system in which these intervals are not mapped in a modulo 2 inverse relation)
>
> Were we to insist that either sharper or flatter is the "better" direction of temperament, consistency would demand that we temper the octave.
>
> As far as polling: cultured opinions are not physiological facts.

I brought up a while ago that I thought that 34-EDO was an ideal
5-limit temperament because everything was just sharp enough,
particularly the fifth. Someone then suggested I just use sharp
octaves.

My cultured opinion is that I like sharp intervals but pure octaves. Sue me.

My tentative physiological explanation is that Setharean dissonance
rules with 2/1, since the second partial is generally much more
prominent than the other partials in a timbre. So even though people
consistently show a preference for sharp octaves when sine waves are
involved, and even though the octave has a pretty low sensitivity to
mistuning, when you start using harmonic timbres its sensitivity to
mistuning suddenly skyrockets.

Or, in short, any beating at the second harmonic will be much more
noticeable than beating at other harmonics, because the second
harmonic itself is much louder than other harmonics. But, as with
anything, I would expect you'd grow to like it if you'd been exposed
culturally to a tempered-octave tuning all your life.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Mon, Apr 4, 2011 at 1:20 AM, lobawad <lobawad@...> wrote:
> >
> > A sharp 3:2 will give a you flat 4:3, a low 5:4 a high 8:5 (assuming an untempered 2 and that we're not using some bizarre system in which these intervals are not mapped in a modulo 2 inverse relation)
> >
> > Were we to insist that either sharper or flatter is the "better" direction of temperament, consistency would demand that we temper the octave.
> >
> > As far as polling: cultured opinions are not physiological facts.
>
> I brought up a while ago that I thought that 34-EDO was an ideal
> 5-limit temperament because everything was just sharp enough,
> particularly the fifth. Someone then suggested I just use sharp
> octaves.

I mentioned the consistent slight sharpness of 6:5, 5:4 and 3:2 several years ago and have worked extensively in 34-edo, an excellent system. But these are not "everything"- 5:3, 8:5 and 4:3 are just as consistently flat.
>
> My cultured opinion is that I like sharp intervals but pure octaves. Sue me.

Why sue you? I was just pointing out discrepancies between theoretical generalizations and real life both mathmatical and artistic.
>
> My tentative physiological explanation is that Setharean dissonance
> rules with 2/1, since the second partial is generally much more
> prominent than the other partials in a timbre. So even though people
> consistently show a preference for sharp octaves when sine waves are
> involved, and even though the octave has a pretty low sensitivity to
> mistuning, when you start using harmonic timbres its sensitivity to
> mistuning suddenly skyrockets.
>
> Or, in short, any beating at the second harmonic will be much more
> noticeable than beating at other harmonics, because the second
> harmonic itself is much louder than other harmonics. But, as with
> anything, I would expect you'd grow to like it if you'd been exposed
> culturally to a tempered-octave tuning all your life.
>
> -Mike
>

On Sun, Apr 3, 2011 at 10:10 PM, Michael <djtrancendance@...> wrote:
>
> Chris>"Middle school science class and up would be fair game. Set up a web site for the kid and let him or her have at it. Got a nephew or niece?"
>
> Well, it's a twisted, but ingenious idea. Picking younger kids virtually guarantees personal bias toward certain intervals (often via "numerology") is eliminated.
>
> I don't have niece/nephew yet...at this rate my up and coming daughter will be in middle school before that happens. But hey...I could always go on a sidetrack and find someone online willing to do it (IE a middle school student looking for class project ideas).

It doesn't, however, ensure that bias towards 12-TET is eliminated.
But it would be a really interesting experiment to do.

-Mike

On 4 April 2011 09:29, Mike Battaglia <battaglia01@...> wrote:

> My tentative physiological explanation is that Setharean dissonance
> rules with 2/1, since the second partial is generally much more
> prominent than the other partials in a timbre. So even though people
> consistently show a preference for sharp octaves when sine waves are
> involved, and even though the octave has a pretty low sensitivity to
> mistuning, when you start using harmonic timbres its sensitivity to
> mistuning suddenly skyrockets.

Even for melodic intervals, the preferred octave size depends on the
timbre. Sine waves are really not representative. People make
mistakes with them because they expect harmonics to be present, and
hear a lower pitch when the harmonics aren't there. We've evolved to
deal with a middling-rich timbre.

It's also a fact that a regular temperament with pure octaves will
give equal and opposite deviations to 3:2 and 4:3. That's a problem
you've evaded.

Graham

On Mon, Apr 4, 2011 at 5:14 AM, Graham Breed <gbreed@...> wrote:
>
> Even for melodic intervals, the preferred octave size depends on the
> timbre. Sine waves are really not representative. People make
> mistakes with them because they expect harmonics to be present, and
> hear a lower pitch when the harmonics aren't there. We've evolved to
> deal with a middling-rich timbre.

People stop preferring sharp octaves with melodic intervals if a
harmonic timbre is used, and they do prefer them sharp if sines are
used?

> It's also a fact that a regular temperament with pure octaves will
> give equal and opposite deviations to 3:2 and 4:3. That's a problem
> you've evaded.

I'm just saying that I so happen to like sharp 3/2's and flat 4/3's...
that's all. Maybe uniformly sharp everything with sharp timbres would
be nice too.

-Mike

On 4 April 2011 13:19, Mike Battaglia <battaglia01@...> wrote:

> People stop preferring sharp octaves with melodic intervals if a
> harmonic timbre is used, and they do prefer them sharp if sines are
> used?

See here:

http://www.mmk.ei.tum.de/persons/ter/top/octstretch.html

"Though the data obtained in the latter study demonstrate that,
grossly, there is a tendency for octave stretch, they also show that,
in an individual ear of an individual listener, the octave deviation W
may at particular frequencies as well be systematically negative."

"Such systematic details of the frequency characteristic of W can
occur only if the experiments are done with sine tones . . . With
harmonic complex tones, the octave stretch is generally smaller than
with corresponding sine tones."

Graham

On Mon, Apr 4, 2011 at 5:35 AM, Graham Breed <gbreed@...> wrote:
>
> See here:
>
> http://www.mmk.ei.tum.de/persons/ter/top/octstretch.html
>
> "Though the data obtained in the latter study demonstrate that,
> grossly, there is a tendency for octave stretch, they also show that,
> in an individual ear of an individual listener, the octave deviation W
> may at particular frequencies as well be systematically negative."
>
> "Such systematic details of the frequency characteristic of W can
> occur only if the experiments are done with sine tones . . . With
> harmonic complex tones, the octave stretch is generally smaller than
> with corresponding sine tones."
>
> Graham

So why was there such doubt over periodicity processing taking place
for melodic intervals if this has always been the case?

-Mike

I made a mistake. On Sunday I suggested that the upper tolerance for tempering a just interval should be +6.776 cents and the lower tolerance should be -27.264 cents. I did a few more listening tests today and I was clearly wrong on the lower tolerance.

I have found in the past that my keyboard behaves strangely sometimes, depending on whether I switch it on before I launch my tuning software program or after I launch it or when I put it into PC mode. My guess is that good quality keyboards don't just play sample sounds simultaneously but rather "mix" the sounds in some way so that they resonate with each other like they would on an acoustic piano. Am I right?

So when I found on Sunday that a -27 cents tempering sounded tolerable perhaps the keyboard wasn't "mixing" (if that's the right word) the notes it was playing.

Anyway I'm back to my old +/-6.776 cents tolerance. By the way, this value is for *me*. I'm putting it out there and people can take it or leave it.

John.

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
>
> Hi Igs,
>
> I've been testing your ideas and got some interesting results. You said that the 5/4 can tolerate more mistuning when you flatten it than if you sharpen it. You were right.
>
> I started with 9/7, a weak interval. I tempered it by +6.5 cents and it still sounded okay. Then I raised it by +8 cents and it sounded sour (so my 6.776 cents "upper" threshold seems good so far). Going the other way (flattening the 9/7) I found that I could flatten the 9/7 by as much as 27 cents and it still sounded "clean". Flattening by 28 cents or more the 9/7 sounded sour.
>
> You said that sometimes the greater the error, the less the discordance and you were right again. I noticed when I was progressively flattening the 9/7 that the beating (discordance) did indeed seem to lessen and the flattened intervals sounded smoother. On the other hand the intervals that were closer to "just" sounded more "resolved". My gut feeling (perhaps because the math looks cuter) is that I would choose "more resolved" over "less beating".
>
> Next I tried 1/1. I tempered the 1/1 by +6.7 cents and it sounded good. Next I raised it by +8 cents and it definitely sounded sour. So again my 6.776 cents "upper" threshold seems good.
>
> Next I tried 2/1. I raised it by 6.7 cents and it was good. +8 cents and it was sour. I flattened 2/1 by -26 cents and the beating was *very* obvious but not "nasty". I tried a -29 cents tempering and the beating began to sound nasty. In this case the -27 cents tolerance is questionable but I wouldn't use a tempered octave in a scale so maybe it doesn't matter.
>
> Next I tried 5/4 and again, flattening the interval by 27 cents, it still sounded "clean". Next I tried -29 cents and I didn't like it.
>
> By coincidence 64/63 is between 27 cents and 28 cents. This is similar to 256/255 (i.e. 2^N/(2^N - 1). So if I had to choose precise values for tempering I would choose +6.776 cents (256/255) for the upper tolerance and -27.264 cents (64/63) for the lower tolerance. I can't prove that these precise values are correct but they look cute and seem reasonable when tested.
>
> John.
>
>
>
> --- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@> wrote:
> >
> > --- In tuning@yahoogroups.com, "john777music" <jfos777@> wrote:
> >
> > > Can you give me an example of an interval where the greater the error the less the
> > > discordance? Sounds wrong to me but I'm curious.
> >
> > The major 3rd is a good example: start from a Just 5/4. Then try 395, 400, 405, 410, 415, and 420 cents. Somewhere between 410 and 420 cents, I hear the discordance diminish, so the peak of discordance is somewhere between 395 and 405. You can also try the perfect fifth: start from 3/2, then flatten to 695, 690, 685, 680, 675, 670, 665. The discordance seems to peak between 680 and 675, and then starts to diminish a bit, bottoming out around 665 cents and then climbing back up as you approach 650 cents.
> >
> > Of course, this effect is most noticeable with timbres where at least the first 16 partials have equal power.
> >
> > -Igs
> >
>

Yes I agree :) 27 cents seemed way to big to me. Personally I work to a tolerance of +-4 cents, but consider much more dissonant JI intervals to be acceptable, such as 121/64 ...

MatC

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
> the lower tolerance should be -27.264 cents. I did a few more listening tests today and I was clearly wrong on the lower tolerance.

Ixtramp>"Yes I agree :) 27 cents seemed way to big to me. Personally I
work to a tolerance of +-4 cents, but consider much more dissonant JI
intervals to be acceptable, such as 121/64 ..."

   4 cents sound to me a virtually undetectable error for just about any ratio.   In contrast, at maximum (IE for intervals like 4/3 that can tolerate a lot of error or small areas between two fairly strong ratios IE between 13/9 and 10/7)...I find 14 cents error can actually be OK.  

   But for worst case scenarios sensitive ratios IE ratios like 22/15 or 11/6, I find even an 8 cent error can can begin to sound intolerably off and I agree around 4 cents or less would be "virtually undetectable". 

---------------------------------------------------

   In summary...it seems we all agree there is NO one error-limit that matches for everything...and I haven't seen anyone argue 7 cents is WAY off the AVERAGE limit: at worst, I've heard it being "a couple of cents off".  Mike B seems interested in finding the "maximum in that can work in the most flexible possible case" error, which I agree is far more than cents, people like Gene seem more interested in where error becomes essentially undetectable (IE 1-3 cents), and John seems interested in an average that's "detectable, but not intolerable across all intervals".

   However...it seems really hard for people on this list to admit...they all have different GOALS when it comes to how they limit error...and thus can have very different answers and yet all be quite justified.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

> Mike B seems interested in finding the "maximum in that can work in the most flexible possible case" error, which I agree is far more than cents, people like Gene seem more interested in where error becomes essentially undetectable (IE 1-3 cents), and John seems interested in an average that's "detectable, but not intolerable across all intervals".

Since I can choose the size of error I like, I am interested not in when error becomes undetectable, which would mean microtemperment, but when it becomes unproblematical. A small error introduces a slight shimmy which makes for a more lively sound. So if you are going to detune things a little anyway, why not detune them in a way which gives some other benefit? Hence, the vast array of temperament possibilities which are not microtemperaments but still quite accurate opens before you.

Maybe this is relevant:

http://apc.psych.umn.edu/Pdfs/Publications/Bernstein_Oxenham_Grouping_Pitch_JASA_2008.pdf

I haven't had a chance to read it yet, but seems interesting. It has to do
with mistuned timbres, so I don't know how much that ties over to mistuned
chords.

-Mike

On Wed, Apr 6, 2011 at 7:35 PM, genewardsmith
<genewardsmith@...>wrote:

>
>
>
> --- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> > Mike B seems interested in finding the "maximum in that can work in the
> most flexible possible case" error, which I agree is far more than cents,
> people like Gene seem more interested in where error becomes essentially
> undetectable (IE 1-3 cents), and John seems interested in an average that's
> "detectable, but not intolerable across all intervals".
>
> Since I can choose the size of error I like, I am interested not in when
> error becomes undetectable, which would mean microtemperment, but when it
> becomes unproblematical. A small error introduces a slight shimmy which
> makes for a more lively sound. So if you are going to detune things a little
> anyway, why not detune them in a way which gives some other benefit? Hence,
> the vast array of temperament possibilities which are not microtemperaments
> but still quite accurate opens before you.
>
>
>